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By-Hand Skills Sheet 6 — Timed Mixed Mock (33 minutes)

Cambridge IGCSE Mathematics 0580 Extended · a construction, bearings, graphs, statistics · 22 marks · Name: ____________________   Date: __________
Sheets 1–5 practise each by-hand skill on its own. A real paper does not: it hands you a construction, then a graph, then a bearings drawing, cold, with a clock running. This sheet is a single 22-mark, 33-minute sitting that mixes every by-hand skill exactly once. Print it, set a timer, and do not open a single model answer until the timer goes.

In the print dialog choose A4 and Scale: 100% — not “Fit to page”. The model answers and this box are hidden when you print, so the paper comes out blank to work on. When the 33 minutes are up, come back to this screen, open the models and mark yourself with the grid at the bottom of the page.

What you need before you start. A pair of compasses that holds its setting, a sharp HB pencil, a 30 cm ruler, a protractor, an eraser — and a timer set to 33 minutes. No calculator: every number on this paper is chosen to be done in your head, which is exactly what non-calculator Paper 2 expects of you.
About the scale bars. Some diagrams carry a printed scale bar marked in units. Set your compasses and take every measurement against that bar, not against a real centimetre ruler — if the printer has scaled the page even slightly, real centimetres will be wrong and the scale bar will still be right. In the exam the diagram is printed accurately and a real ruler is correct, so this is the one place this sheet differs from a real paper.

Exam conditions  ·  read before the timer starts

Treat this exactly like the real thing. The habits are the point.
  1. Print first, then start the timer. 33 minutes, one sitting, no pauses. About 7 minutes for each 5-mark question and 3 for Q1, which leaves 2 minutes to check.
  2. No calculator anywhere on this paper.
  3. Never rub out a construction arc. The arcs are the method marks.
  4. Rule every straight line — tangents, triangle sides, lines of best fit, bearings legs.
  5. Show where you read every value from: dashed lines to the axis, a gradient triangle, a dashed measuring line. These carry marks of their own.
  6. When time is up, stop, then mark yourself honestly against the models — mark by mark, using the grid at the bottom — and read the guidance under it.

The paper  ·  5 questions · 22 marks · 33 minutes

Every by-hand skill appears exactly once. Answer on this sheet.
Q1 The line AB is 8 units long. Using a ruler and compasses only, construct triangle ABC with AC = 6 units and BC = 7 units, with C above AB. Set your compasses against the scale bar. [2]
C is where an arc of radius 6 from A crosses an arc of radius 7 from B.
A B 8 units 0 4 units
How the marks are awarded B1 the two locating arcs, radius 6 from A and radius 7 from B, still visible; B1 triangle ruled through their crossing. A triangle made with a protractor, or with no arcs, scores the drawing marks as 0.
A B 8 units 0 4 units C 6 7 leave every arc in
  • Marking points
  • Compasses at 6 units (set on the scale bar) arc from A; at 7 units arc from B. C is the crossing. Both arcs stay on the page.
  • Sides AC and BC are ruled through C. Tolerance is about 2 mm at each vertex — a wobbly compass setting eats it instantly.
Q2 (a) Complete the table of values for y = x² − 2x − 3. [1]  (b) Plot all seven points on the grid and draw a smooth curve through them. [2]  (c) Draw the tangent to the curve at x = 3 and use it to estimate the gradient of the curve there. [2] [5]
Squaring a negative gives a positive: (−1)² = 1. And a parabola is symmetric — if your table has no matching pairs, a value is wrong.
x−2−101234
y5 −3−4 0 
-2 -1 1 2 3 4 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
How the marks are awarded 1 for all three table values (−1 → 0, 2 → −3, 4 → 5). P1 for all seven points plotted within half a small square; C1 for one smooth curve through them — straight-line segments or a flat-bottomed U lose it. T1 for a ruled tangent touching the curve at (3, 0) without cutting it; A1 for a gradient from a large triangle on that tangent: the true value is 4, accept 3 to 5.
-2 -1 1 2 3 4 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y 2 8 tangent at x = 3 gradient ≈ 8 ÷ 2 = 4
  • Marking points
  • Table: x = −1 gives 1 + 2 − 3 = 0; x = 2 gives 4 − 4 − 3 = −3; x = 4 gives 16 − 8 − 3 = 5. Symmetric about x = 1, as a parabola must be.
  • Plot with small neat crosses. The turning point sits at (1, −4) — the curve turns at a point, it is never flat.
  • One smooth stroke through all seven crosses. If it looks hairy, erase and redraw the whole curve, not a patch of it.
  • The tangent is ruled, touches only at (3, 0), and is extended well past it — a long line makes the triangle big and the estimate honest.
  • Gradient triangle: from (2, −4) to (4, 4) the rise is 8 and the run is 2, so gradient ≈ 8 ÷ 2 = 4. Anything from 3 to 5 with a visible triangle earns the mark.
  • The dashed triangle is not decoration — with no working shown, a bare “4” can score zero on a read-from-the-graph question.
Q3 60 students were timed solving a puzzle. (a) Complete the cumulative frequency row. [1]  (b) Plot the cumulative frequency curve on the grid. [2]  (c) Use your curve to estimate the median and the interquartile range. [2] [5]
Cumulative frequency is plotted at the upper end of each class — 6 students had finished by 10 minutes, not at 5.
time t (min)0 < t ≤ 1010 < t ≤ 2020 < t ≤ 3030 < t ≤ 4040 < t ≤ 50
frequency61420146
cumulative frequency620 54 
0 10 20 30 40 50 60 10 20 30 40 50 60 0 time t (minutes) cumulative frequency
How the marks are awarded 1 for both missing values (40 and 60). P1 for points at the upper class boundaries — (10, 6), (20, 20), (30, 40), (40, 54), (50, 60) — plotting at midpoints shifts every answer and is the single most common error; C1 for a smooth increasing curve through them from (0, 0). B1 median ≈ 25 (accept 24 to 26); B1 IQR ≈ 17 (accept 15 to 19), with dashed read-off lines at cumulative frequencies 15, 30 and 45.
0 10 20 30 40 50 60 10 20 30 40 50 60 0 time t (minutes) cumulative frequency median ≈ 25 LQ ≈ 16.5 UQ ≈ 33.5 IQR ≈ 33.5 − 16.5 = 17
  • Marking points
  • Cumulative frequency just keeps adding: 6, 20, then 20 + 20 = 40, 54, then 54 + 6 = 60 — and the last value must equal the total, 60. That is your built-in check.
  • Points go at the upper boundary of each class, and the curve starts at (0, 0) because nobody had finished at time 0.
  • A cumulative frequency curve can never come back down. If yours dips, a point is misplotted.
  • Median: 60 students, so read across from cumulative frequency 30 → t ≈ 25 minutes.
  • Quartiles: read across from 15 and 45 → LQ ≈ 16.5, UQ ≈ 33.5, so IQR ≈ 33.5 − 16.5 = 17 minutes.
  • The dashed lines are the working. Reading at frequencies 15/30/45 of the data values instead of the cumulative axis is the other classic slip — always start on the vertical axis.
Q4 A ship leaves the harbour H and sails 5 km on a bearing of 070° to a buoy P. It then sails 4 km on a bearing of 160° to a lighthouse Q. (a) Using the scale bar, make an accurate scale drawing of the journey. [3]   (b) By measuring your drawing, find the distance HQ in km and the bearing of H from Q. [2] [5]
A bearing is measured clockwise from north, three figures. Before you measure 160° at P, draw a fresh north line at P — the protractor centre goes where the ship is.
H N 0 4 km
How the marks are awarded B1 leg HP: bearing 070° (±2°) and length 5 km (±0.1 km on the scale bar). B1 a new north line drawn at P and leg PQ at 160° (±2°), 4 km long — measuring the second bearing from H’s north line instead of P’s is the classic error. B1 a complete, labelled drawing (H, P, Q, north lines). B1 HQ ≈ 6.4 km (accept 6.2 to 6.6); B1 bearing of H from Q ≈ 289° (accept 286 to 292).
H N 0 4 km N N P Q 070° 160° 5 km 4 km HQ ≈ 6.4 km 289°
  • Marking points
  • Protractor centred on H, 0 on the north line, measure 70° clockwise; rule the leg and set 5 km with compasses against the scale bar.
  • New north line at P, then 160° clockwise from it. The two legs make a right angle at P (160 − 70 = 90) — your drawing should visibly show one.
  • Because of that right angle, HQ is exactly √(5² + 4²) = √41 ≈ 6.40 km — so a careful drawing measures 6.3–6.5.
  • Bearing of H from Q: protractor centred on Q, north line at Q, clockwise round to the direction of H ≈ 289°. It is a reflex bearing; if you measured the small angle 71° on the other side, subtract from 360.
  • Check without a protractor: the bearing of Q from H measures ≈ 109°, and a back bearing differs by 180: 109 + 180 = 289 ✓.
  • Three figures always: 070°, never 70°.
Q5 The table shows the midday temperature and the number of cups of hot chocolate a café sold on eight days. (a) Plot a scatter diagram. [2]   (b) Draw a ruled line of best fit. [1]  (c) Use your line to estimate sales on a day when the midday temperature is 10 °C. [1]  (d) The owner wants to use the line to predict sales at 25 °C. In one sentence, say whether that prediction is reliable, and why. [1] [5]
Work out the mean of each row first — both come out exactly in your head (the temperatures sum to 64, the cups to 208) — and put your ruler through that point.
temperature, t (°C)24579111214
cups sold, c3834332824201813
0 2 4 6 8 10 12 14 16 5 10 15 20 25 30 35 40 midday temperature (°C) cups of hot chocolate sold
How the marks are awarded P2 for all eight points plotted within half a small square (P1 for six or seven). B1 for one ruled line with the points balanced either side, passing through the mean point (8, 26). B1 for an estimate of about 22 cups (accept 20 to 24) with dashed read-off lines shown. B1 for the sentence: not reliable, because 25 °C is far outside the range of the data — that is extrapolation.
0 2 4 6 8 10 12 14 16 5 10 15 20 25 30 35 40 midday temperature (°C) cups of hot chocolate sold mean point (8, 26) about 22
  • Marking points
  • Mean temperature = 64 ÷ 8 = 8. Mean cups = 208 ÷ 8 = 26. Plot (8, 26) and put your ruler on it.
  • One ruled line, drawn right across the data, roughly four points each side. Not a curve, not freehand, not joined dot-to-dot.
  • The correlation is strong negative — colder days sell more hot chocolate. Use both words if asked to describe it.
  • Reading at t = 10 gives about 22 cups. Dashed lines up from 10 and across to the axis are the working, and carry the mark.
  • At 25 °C the line predicts 26 − 2 × 17 ≈ −9 cups — a negative number of drinks. That absurdity is the answer: 25 °C is far beyond the data (extrapolation), so the prediction is not reliable.
  • One sentence with the word outside the data range (or extrapolation) earns it; “no, because it might be wrong” does not.

Mark yourself  ·  out of 22

When the 33 minutes are up: open each model, award your marks one by one against the marking points, and fill in this grid in pencil. Be exact, not kind — a triangle with no construction arcs is 0, not “nearly 2”.
QuestionSkillMarks availableMarks earned
Q1Triangle construction2 
Q2Quadratic curve, tangent, gradient5 
Q3Cumulative frequency, median, IQR5 
Q4Bearings scale drawing5 
Q5Scatter, best fit, reliability5 
Total 22 
  1. 18 or more: you are on A pace for by-hand skills under time pressure. Keep the edge with an occasional exercise from Sheets 1–5 — these skills rust quietly.
  2. 13 to 17: the skills are there but leaking marks. Look at which marks went — almost always missing arcs, missing dashed working, or a second bearing measured from the wrong north line. Redo just the questions below full marks, untimed, this week.
  3. Below 13: no verdict on you — just on today. Reprint the whole sheet and sit it again in one week: first untimed with Sheets 1–5 open, then a few days later timed and closed. Second sittings of this sheet routinely jump 6–7 marks.
Cambridge IGCSE Mathematics 0580 Extended · by-hand skills · Sheet 6: timed mixed mock · 5 questions · 22 marks · 33 minutes · triangle construction, bearings, graphs and tangents, cumulative frequency, scatter diagrams · Print at 100% on A4 and complete in pencil.